We have now derived the fundamental models for
cables and active membranes and thus should be able to produce models
of individual neurons and banched neuronal structures such as
dendrites and axons. Networks of neurons communicate in a variety of
means but by far the most common is through synapses. Synapses are
exceeding complex and involve a large number of biochemical steps, can
be modified and modulated by external substances, and are capable of
undergoing slow adaptation and facilitation. Nevertheless, we will
blunder on in order to describe ways in which one might model them;
for without synapse, we cannot model networks and without networks, we
are nothing. Indeed, to paraphrase Dorothy Parker, you can put a
cell in culture but you can't make it think. The first step in this
process of communication is the action potential;
abbreviated as ``AP.'' The action potential occurs when we combine the
active membrane theory of ``channeling'' with the passive cable
theory. The compartmental model is then:

where Iact contains all of the active channels with conductances
given in terms of density per unit area and l is the length of the
compartment and d is the diameter of the cylinder. Dividing by
and letting to approximate a continuous axon, we
obtain:

which is the model for the continuous axon originally modeled by HH.
One can numerically simulate this partial differential equation and
it is found that for each set of parameters, there is a solution that
is a travelling wave. That is, there is a solution that keeps the
same spatial profile translated in time. Specifically,
V(x,t)=U(x-ct) where U is a particular function and c is the
velocity of the wave. (See illustration of the wave.) A natural
question to ask is how does the velocity depend on say, the geometry
or the coupling? This is easily seen from the above. The term d
multiplies two spatial derivatives of V, thus, the velocity depends
on d as the square-root. That is, quadrupling the diameter doubles
the velocity. Similarly quadrupling the intra-axonal resistance halves
the velocity. The rigorous existence of traveling waves to the HH
equations was independently proven in the 70's by Stuart Hastings and
Gail Carpenter.

The AP provides the means of communicating between neurons by
activating the axon and causing a signal to travel unattenuated down
the cable. It is possible to block this by various means. For
example, if the diameter of the axon increases drastically (say at a
branch point) then the impuls can be blocked. Since there is no
intrinsic anisotropy in the axon, it is possible to initiate an
impulse any where and it will propagate outward (see Figure.)
Because of the refractory period following the AP, the conducting
medium behind the AP is very hyperpolarized and remains so until the
sodium-potassium pump is able to rebalance the ionic concentrations.
As a consequence of this, two propagating action potentials that
collide, annihilate, unlike more common physical waves such as boles or
light waves which pass through each other unchanged. (Note that for
linear waves, this is a trivial observation, but for boles this
requires some amount of mathematics.)

Figure 3:
Voltage contour of an action potential in space-time

Repeated stimuli can have somewhat complicated effects. The
simplest is to produce a train of waves that are equally spaced dowm
the axon. If we let denote the velocity of the waves and
T their period, then there is a relationship between
and T called the dispersion relation. Any nonlinear
medium capable of producing waves has such a relationship. You have
probably observed it in water; waves with different magnitudes have
different velocities and thus tend to disperse through the medium.
Generally, the higher the temporal frequency, the smaller is the
amplitude and velocity. The fastest wave is the one with zero
frequency; the solitary pulse. The quantity, has
dimensions of 1/dist and is called the wavenumber or spatial
frequency of the waves. Nonmonotone dispersion relations have
profound consequences for the spacing of waves and it is possible to
get doublets and other complicated spacings of impulses. The
mathematical analysis of this type of phenomena has led to some
striking insights into propagation of the lowly action potential.

In spite of all this interesting behavior, when it comes time to model
networks of neurons, most people ignore it and model the action
potential via a conduction delay. The range of velocities are from
2-10 meters/second for a typical impulse, the faster occuring in
myelinated axons. In a small piece of cortical tissue a millimeter
on the side, the maximum of these delays is less than a millisecond
so that for purposes of modeling we will generally neglect them.
However, in parts of the olfactory cortex, there appear to be
instances where conduction delays are important.